Interactive bicriterion solution method and its application to critical path method problems

An interactive solution method is developed for bicriterion mathematical programming (BCMP) problems. The new method, called the dichotomous bicriterion mathematical programming (DBCMP) method, combines Tchebycheff theory and the existing paired comparison method (PCM). The DBCMP method is then compared with the PCM method based on critical path method problems with two conflicting objectives: minimizing the total crashing cost and minimizing the total project completion time. The extension of the DBCMP method to BCMP problems with multiple decision makers is also discussed.     

On the application of the asymptotic method of global instability in aeroelasticity problems

The asymptotic method of global instability developed by A.G. Kulikovskii is an effective tool for determining the eigenfrequencies and stability boundary of one-dimensional or multidimensional systems of sufficiently large finite length. The effectiveness of the method was demonstrated on a number of one-dimensional problems; and since the mid-2000s, this method has been used in aeroelasticity problems, which are not strictly one-dimensional: such is only the elastic part of the problem, while the gas flow occupies an unbounded domain. In the present study, the eigenfrequencies and stability boundaries predicted by the method of global instability are compared with the results of direct calculation of the spectra of the corresponding problems. The size of systems is determined starting from which the method makes a quantitatively correct prediction for the stability boundary.     

Approximate solution to the diffusion equation and its application to seepage-related problems

Effective dam safety procedures require pore pressure measurements to be interpreted as soon as possible after readings have been taken. Direct interpretations based on partial differential equations are not appropriate. Here, we took a different approach which consisted in applying the impulse response function concept to a dam site. Consequently, two models were designed so that their parameters matched the temporal moments of Green’s function of the associated parabolic problem. These models were compared with closed-form solutions. The first model was the exponential decay model, which gave good results, but did not account for high loading harmonics. The second model, based on the fundamental solution of a parabolic problem, was surprisingly accurate. As an example problem we considered monitoring data obtained from a zoned earth dam. The results show that the main aspects of the processes reflected in most cell recordings can be described in a linear framework, and that they are accounted for by both models.     

双边配给问题的Shapley解及其在博物馆通票问题中的应用

双边配给问题描述了现实生活中一类带有二部图结构的稀缺资源配置问题, 例如, 在自然灾害期间救援物资的配给; 电力和天然气等自然资源按需分配; 高校引进人才调配等。本文通过求解线性规划, 并从联盟边际贡献的角度出发定义了双边配给问题的一个Shapley解。之后, 通过合作对策模型和解的公理化方法说明新解的合理性。首先, 建立双边配给问题的合作对策模型, 论证了新解与双边配给合作对策的Shapley值一致; 其次, 证明了Shapley解是唯一满足优先一致性的有效配给方案。最后, 将Shapley解应用于博物馆通票问题的研究, 探讨了博物馆合作制定通票后所得单票和通票收益的分配方式。     

双边配给问题的Shapley解及其在博物馆通票问题中的应用

双边配给问题描述了现实生活中一类带有二部图结构的稀缺资源配置问题, 例如, 在自然灾害期间救援物资的配给; 电力和天然气等自然资源按需分配; 高校引进人才调配等。本文通过求解线性规划, 并从联盟边际贡献的角度出发定义了双边配给问题的一个Shapley解。之后, 通过合作对策模型和解的公理化方法说明新解的合理性。首先, 建立双边配给问题的合作对策模型, 论证了新解与双边配给合作对策的Shapley值一致; 其次, 证明了Shapley解是唯一满足优先一致性的有效配给方案。最后, 将Shapley解应用于博物馆通票问题的研究, 探讨了博物馆合作制定通票后所得单票和通票收益的分配方式。     

The method of Iterated Defect-Correction and its application to two-point boundary value problems

Summary In Part II of this paper we present a rigorous analysis of the Iterated Defect-Correction — applied to two-point boundary value problems — which was introduced in Part I of this paper [1]. A complete proof of Theorem 5. 1 of [1] is given.     

On the solution set of convex problems and its numerical application

The solution set of a convex problem is enough interested. In this paper we have a discussion to determine it. The conditions and proofs in this note essentially differs from the previous well-known works. We also have a new result determining the solution of minimum norm for quadratic problems.     

On a computational method for the second fundamental tensor and its application to bifurcation problems

Summary An algorithm is presented for the computation of the second fundamental tensorV of a Riemannian submanifoldM ofR ⁿ . FromV the riemann curvature tensor ofM is easily obtained. Moreover,V has a close relation to the second derivative of certain functionals onM which, in turn, provides a powerful new tool for the computational determination of multiple bifurcation directions. Frequently, in applications, thed-dimensional manifoldM is defined implicitly as the zero set of a submersionF onR ⁿ . In this case, the principal cost of the algorithm for computingV(p) at a given pointpM involves only the decomposition of the JacobianDF(p) ofF atp and the projection ofd(d+1) neighboring points ontoM by means of a local iterative process usingDF(p). Several numerical examples are given which show the efficiency and dependability of the method.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayThis work was in part supported by the National Science Foundation (DCR-8309926) and the Office of Naval Research (N-00014-80-C09455). The second author began some of the work while visiting the University of Heidelberg/Germany as an Alexander von Humboldt Senior U.S. Scientist     

Predictor homotopy analysis method and its application to some nonlinear problems

The purpose of this paper is to present a kind of analytical method so-called Predictor homotopy analysis method (PHAM) to predict the multiplicity of the solutions of nonlinear differential equations with boundary conditions. This method is very useful especially for those boundary value problems which admit multiple solutions and furthermore is capable to calculate all branches of the solutions simultaneously. As illustrative examples, the method is checked by the model of mixed convection flows in a vertical channel and a nonlinear model arising in heat transfer which both admit multiple (dual) solutions.     

The convergence of the perturbed Newton method and its application for ill-conditioned problems

Iterative methods, such as Newton’s, behave poorly when solving ill-conditioned problems: they become slow (first order), and decrease their accuracy. In this paper we analyze deeply and widely the convergence of a modified Newton method, which we call perturbed Newton, in order to overcome the usual disadvantages Newton’s one presents. The basic point of this method is the dependence of a parameter affording a degree of freedom that introduces regularization. Choices for that parameter are proposed. The theoretical analysis will be illustrated through examples.     

The penetration function and its application to microscale problems

The penetration function measures the effect of the boundary data on the energy of the solution of a second order linear elliptic PDE taken over an interior subdomain. Here the coefficients of the PDE are functions of position and often represent the material properties of non homogeneous media with microstructure. The penetration function is used to assess the accuracy of global-local approaches for recovering local solution features from coarse grained solutions such as those delivered by homogenization theory. AMS subject classification (2000)  65N15, 78M40     

A matroid algorithm and its application to the efficient solution of two optimization problems on graphs

We address the problem of finding a minimum weight baseB of a matroid when, in addition, each element of the matroid is colored with one ofm colors and there are upper and lower bound restrictions on the number of elements ofB with colori, fori = 1, 2,,m. This problem is a special case of matroid intersection. We present an algorithm that exploits the special structure, and we apply it to two optimization problems on graphs. When applied to the weighted bipartite matching problem, our algorithm has complexity O(|EV|+|V| ²log|V|). HereV denotes the node set of the underlying bipartite graph, andE denotes its edge set. The second application is defined on a general connected graphG = (V,E) whose edges have a weight and a color. One seeks a minimum weight spanning tree with upper and lower bound restrictions on the number of edges with colori in the tree, for eachi. Our algorithm for this problem has complexity O(|EV|+m ² |V|+ m|V| ²). A special case of this constrained spanning tree problem occurs whenV ^* is a set of pairwise nonadjacent nodes ofG. One must find a minimum weight spanning tree with upper and lower bound restrictions on the degree of each node ofV ^*. Then the complexity of our algorithm is O(|VE|+|V ^* V| ²). Finally, we discuss a new relaxation of the traveling salesman problem.This report was supported in part by NSF grant ECS 8601660.     

A new approach to numerical solution of fixed-point problems and its application to delay differential equations

In this paper we consider a certain approximation of fixed-points of a continuous operator A mapping the metric space into itself by means of finite dimensional ε(h)-fixed-points of A. These finite dimensional functions are obtained from functions defined on discrete space grid points (related to a parameter h→0) by applying suitably chosen extension operators p_h. A theorem specifying necessary and sufficient conditions for existence of fixed-points of A in terms of ε(h)-fixed-points of A is given. A corollary which follows the theorem yields an approximate method for a fixed-point problem and determines conditions for its convergence. An example of application of the obtained general results to numerical solving of boundary value problems for delay differential equations is provided.Numerical experiments carried out on three examples of boundary value problems for second order delay differential equations show that the proposed approach produces much more accurate results than many other numerical methods when applied to the same examples.     

The application of the Ehrlich-Aberth method to structured polynomial eigenvalue problems

An algorithm for the application of the Ehrlich-Aberth method to polynomial eigenvalue problems (PEPs) is proposed. The computational complexity of the algorithm is only quadratic with respect to the degree of the polynomial, where customary matrix methods have cubic complexity. For the case of some structured PEPs a strategy is given in order to exploit the structure and obtain the induced coupling in the spectrum. Numerical experiments are provided for the particular case of even/odd PEPs. This communication is based on joint work with Dario A. Bini and Luca Gemignani. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

One class of separable optimization problems: solution method,application

Convexity and generalized convexity play a central role in mathematical economics and optimization theory. So, the research on criteria for convexity or generalized convexity is one of the most important aspects in mathematical programming, in order to characterize the solutions set. Many efforts have been made in the few last years to weaken the convexity notions. In this article, taking in mind Craven's notion of K-invexity function (when K is a cone in ? ⁿ ) and Martin's notion of Karush–Kuhn–Tucker invexity (hereafter KKT-invexity), we define a new notion of generalized convexity that is both necessary and sufficient to ensure every KKT point is a global optimum for programming problems with conic constraints. This new definition is a generalization of KKT-invexity concept given by Martin and K-invexity function given by Craven. Moreover, it is the weakest to characterize the set of optimal solutions. The notions and results that exist in the literature up to now are particular instances of the ones presented here.     

A feasible decomposition method for constrained equations and its application to complementarity problems

In this paper, by analyzing the propositions of solution of the convex quadratic programming with nonnegative constraints, we propose a feasible decomposition method for constrained equations. Under mild conditions, the global convergence can be obtained. The method is applied to the complementary problems. Numerical results are also given to show the efficiency of the proposed method.     

The global element method and its extension to parabolic problems

This article describes the derivation of the global element method, originally proposed by Delves and Hall, using the classical Euler–Lagrange theory. The method is then extended so as to be applicable to parabolic partial differential equations. The main advantages of the method, the ease of adjusting its accuracy and applying it to problems with discontinuous coefficients and point singularities, are also discussed.